Difficult Integration by parts

Evaluate $\displaystyle \int \frac{sin(\frac{1}{x})}{x^3} \, dx$

For an integral like this one, where the sine function is combined with a power of x in the denominator, it's helpful to recognize when substitution is the best method. In this case, the function inside the sine, one over x, suggests that we should use a substitution to simplify the integral.

The substitution will allow you to rewrite the integral in a way that simplifies both the trigonometric and algebraic components. After substituting and adjusting the differential accordingly, you’ll find that the integral becomes easier to solve. Once the integration is complete, you substitute back to return to the original variable, giving the final solution.

This type of problem shows how substitution can be an effective strategy for integrals involving both trigonometric functions and powers of x, especially when the argument of the trigonometric function is itself a more complex expression like one over x. It turns what might initially seem like a complicated integral into a more manageable one by reducing the complexity of the terms involved.

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